Solving Inequalities

One topic that has nearly vanished from my teaching is the solving of inequalities “by hand”. There are several reasons for that choice:

The techniques are difficult to teach and difficult to learn, because they are so close to the ones for the solving of linear equations, but differ in one crucial case.

It is difficult to understand the reasons for that exception.

Much understanding about functions is conferred by using Cartesian graphs instead (graph each side of the inequality, and you can readily see the values of x that make one or the other side greater than the other.)

The graphical / electronic approach is not limited to linear inequalities: it works in pretty much any case one cares to explore.

If you teach for understanding, you need to teach fewer topics, and prioritize. Given the fact that epsilon-delta proofs are no longer part of anyone’s expectations for high school, the manipulation of inequalities is less of a priority than it once was. (And even more so, the solving of inequalities involving absolute values!)

Amy L. Nebesniak, in “Effective Instruction” (The Mathematics Teacher, Vol 106, No. 5, Dec 2012/Jan 2013) argues that if you want to teach students to manipulate linear inequalities, the key is to address the second point on the above list: understanding. As it turns out, the approach she recommends is quite compatible with the use of function diagrams — enough so that I added a section to the Function Diagrams page on my Web site (and a shout-out to Amy in the bibliography.)

There are many reasons to use function diagrams in your teaching at all levels of secondary school. If you teach solving inequalities, you now have one more reason.